Price of Anarchy and Stability Non-atomic games: infinite # players with infinitesimal demand. PoA for non-atomic games with latencies in class D PoA = α ( D ) [Rough03] 4 / 3 for linear latencies, Θ( p ln p ) for polynomials of degree p . PoA for atomic games with unsplittable demands Linear latencies: PoA = 2 . 5 [AzarAweEpst05], [ChristKouts05], [AlandDGMS06] Polynomial latencies of degree p : PoA = p Θ( p ) Parallel links : PoA = α ( D ) [L¨ uckMMR04], [CaragKaklKanel07], [Fot07] Extension-parallel networks : PoA = α ( D ) [Fotakis08] Price of Stability √ Linear latencies: PoS = 1 + 3 / 3 [ChristKouts05], [CaragKaklamKanel07] s − t networks : PoS = α ( D ) [Fotakis08] Dimitris Fotakis Congestion Games and Price of Anarchy
Reducing the Price of Anarchy Network Design Detection (and elimination) of Braess’s paradox [Rough01] Difficult if the network is operational, computationally hard Tractable for several interesting cases [FotaKapoSpir 09] Dimitris Fotakis Congestion Games and Price of Anarchy
Reducing the Price of Anarchy Network Design Detection (and elimination) of Braess’s paradox [Rough01] Difficult if the network is operational, computationally hard Tractable for several interesting cases [FotaKapoSpir 09] Coordination mechanisms Modify the players’ costs [ChristKoutsNanav 04] Scheduling [ImmorLMS 05],[AzarJainMir 08], [Caragiannis 09] Connection games : priority cost-sharing better than fair [CRV08] Significant improvement. Fairness and implementation issues. Dimitris Fotakis Congestion Games and Price of Anarchy
Reducing the Price of Anarchy Stackelberg routing Exploit the presence of coordinated players [KorLazOrd 97] Improvement depends on the fraction of coordinated players. No system modifications are required. Dimitris Fotakis Congestion Games and Price of Anarchy
Reducing the Price of Anarchy Stackelberg routing Exploit the presence of coordinated players [KorLazOrd 97] Improvement depends on the fraction of coordinated players. No system modifications are required. Resource pricing Introduce economic disincentives (refundable tolls). Tolls increase players’ disutility, large tolls may be required. Tolls known to enforce optimal configuration for non-atomic games. Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Routing Model Both selfish and coordinated players are present. Leader determines paths of coordinated players to optimize performance. Selfish players ( followers ) seek to minimize their own latency and reach a pure Nash equilibrium. Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Routing Model Both selfish and coordinated players are present. Leader determines paths of coordinated players to optimize performance. Selfish players ( followers ) seek to minimize their own latency and reach a pure Nash equilibrium. Stackelberg Strategy Algorithm allocating a path to each coordinated player. Objective: lead the selfish players to a good PNE. Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Γ( N , G ( V , E ) , ( d e ) e ∈ E ) : k coordinated and n − k selfish players. Fraction of coordinated players : γ = k / n Optimal configuration o = ( o 1 , . . . , o n ) Poly-time for symmetric network games with convex latencies. Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Γ( N , G ( V , E ) , ( d e ) e ∈ E ) : k coordinated and n − k selfish players. Fraction of coordinated players : γ = k / n Optimal configuration o = ( o 1 , . . . , o n ) Poly-time for symmetric network games with convex latencies. Coordinated players are assigned to k optimal paths . Stackelberg strategy selects L ⊆ N , | L | = k = γ n Stackelberg configuration s ( L ) = ( o i ) i ∈ L Stackelberg congestion s e ( L ) = |{ i ∈ L : e ∈ o i }| Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Γ( N , G ( V , E ) , ( d e ) e ∈ E ) : k coordinated and n − k selfish players. Fraction of coordinated players : γ = k / n Optimal configuration o = ( o 1 , . . . , o n ) Poly-time for symmetric network games with convex latencies. Coordinated players are assigned to k optimal paths . Stackelberg strategy selects L ⊆ N , | L | = k = γ n Stackelberg configuration s ( L ) = ( o i ) i ∈ L Stackelberg congestion s e ( L ) = |{ i ∈ L : e ∈ o i }| Game ˜ Γ L ( N \ L , G ( V , E ) , (˜ d e ) e ∈ E ) , with ˜ d e ( x ) = d e ( x + s e ( L )) Selfish players : worst PNE σ ( L ) of maximum C ( σ ( L ) + s ( L )) Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Γ( N , G ( V , E ) , ( d e ) e ∈ E ) : k coordinated and n − k selfish players. Fraction of coordinated players : γ = k / n Optimal configuration o = ( o 1 , . . . , o n ) Poly-time for symmetric network games with convex latencies. Coordinated players are assigned to k optimal paths . Stackelberg strategy selects L ⊆ N , | L | = k = γ n Stackelberg configuration s ( L ) = ( o i ) i ∈ L Stackelberg congestion s e ( L ) = |{ i ∈ L : e ∈ o i }| Game ˜ Γ L ( N \ L , G ( V , E ) , (˜ d e ) e ∈ E ) , with ˜ d e ( x ) = d e ( x + s e ( L )) Selfish players : worst PNE σ ( L ) of maximum C ( σ ( L ) + s ( L )) Price of Anarchy PoA Γ Str ( γ ) = max C ( σ ( L ) + s ( L )) / C ( o ) σ ( L ) ∈ PNE (˜ Γ L ) Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Γ( N , G ( V , E ) , ( d e ) e ∈ E ) : k coordinated and n − k selfish players. Fraction of coordinated players : γ = k / n Optimal configuration o = ( o 1 , . . . , o n ) Poly-time for symmetric network games with convex latencies. Coordinated players are assigned to k optimal paths . Stackelberg strategy selects L ⊆ N , | L | = k = γ n Stackelberg configuration s ( L ) = ( o i ) i ∈ L Stackelberg congestion s e ( L ) = |{ i ∈ L : e ∈ o i }| Game ˜ Γ L ( N \ L , G ( V , E ) , (˜ d e ) e ∈ E ) , with ˜ d e ( x ) = d e ( x + s e ( L )) Selfish players : worst PNE σ ( L ) of maximum C ( σ ( L ) + s ( L )) Price of Anarchy PoA G Str ( γ ) = sup max C ( σ ( L ) + s ( L )) / C ( o ) σ ( L ) ∈ PNE (˜ Γ L ) Γ ∈G Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Computing the best strategy is NP -complete even for parallel links and linear latencies [Roughgarden 02] . Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Computing the best strategy is NP -complete even for parallel links and linear latencies [Roughgarden 02] . Largest Latency First – LLF [Rough02] Coordinated players to optimal paths of largest latency. If c 1 ( o ) ≥ · · · ≥ c n ( o ) , then L = { 1 , . . . , k } . Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Computing the best strategy is NP -complete even for parallel links and linear latencies [Roughgarden 02] . Largest Latency First – LLF [Rough02] Coordinated players to optimal paths of largest latency. If c 1 ( o ) ≥ · · · ≥ c n ( o ) , then L = { 1 , . . . , k } . Scale [Rough02] � n � Random set L , | L | = k , with probability 1 / . k Each resource e gets γ o e coordinated players on expectation. Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies Computing the best strategy is NP -complete even for parallel links and linear latencies [Roughgarden 02] . Largest Latency First – LLF [Rough02] Coordinated players to optimal paths of largest latency. If c 1 ( o ) ≥ · · · ≥ c n ( o ) , then L = { 1 , . . . , k } . Scale [Rough02] � n � Random set L , | L | = k , with probability 1 / . k Each resource e gets γ o e coordinated players on expectation. Cover Atomic games when k large enough (e.g. k ≥ m ) . Integer λ : each e gets ≥ min { λ, o e } coordinated players. L is computed greedily so that min { λ, o e } ≤ s e ( L ) ≤ o e for all e . Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal Worst PNE x 11 18 Total lat.: 255 Total lat.: 324 2 0 2 x +16 2 x +16 t 2 t 0 t 18 s 18 s 18 s players players players 2 x +16 1 0 1 0 2 x +16 2 x +16 1 0 Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal Worst PNE x 11 18 Total lat.: 255 Total lat.: 324 2 0 2 x +16 2 x +16 t 2 t 0 t 18 s 18 s 18 s players players players 2 x +16 1 0 1 0 2 x +16 2 x +16 1 0 LLF, 6 players 12 Total lat.: 260 2 2 s t 18 players 1 1 0 Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal Worst PNE x 11 18 Total lat.: 255 Total lat.: 324 2 0 2 x +16 2 x +16 t 2 t 0 t 18 s 18 s 18 s players players players 2 x +16 1 0 1 0 2 x +16 2 x +16 1 0 LLF, 6 players Cover, 6 players 12 1 + 12 Total lat.: 260 Total lat.: 259 2 1 2 1 s t s t 18 18 players players 1 1 1 1 0 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal Worst PNE x 11 18 Total lat.: 255 Total lat.: 324 2 0 2 x +16 2 x +16 t 2 t 0 t 18 s 18 s 18 s players players players 2 x +16 1 0 1 0 2 x +16 2 x +16 1 0 LLF, 6 players Cover, 6 players 12 1 + 12 Total lat.: 260 Total lat.: 259 2 1 2 1 s t s t 18 18 players players 1 1 1 1 0 1 Scale : I E [ C ( s + σ )] ≥ 284 . 98 − With probability ≥ 0 . 572, C ( s + σ ) ≥ 292 − With probability ≥ 0 . 883, C ( s + σ ) ≥ 279 − With probability ≥ 0 . 987, C ( s + σ ) ≥ 268 Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal Worst PNE x 11 18 Total lat.: 255 Total lat.: 324 2 0 2 x +16 2 x +16 t 2 t 0 t 18 s 18 s 18 s players players players 2 x +16 1 0 1 0 2 x +16 2 x +16 1 0 Optimal strategy LLF, 6 players Cover, 6 players 6 players 12 1 + 12 12 Total lat.: 260 Total lat.: 259 Total lat: 256 2 1 2 2 1 1 s t s t t 18 18 18 s players players players 1 1 1 1 1 1 0 1 1 Scale : I E [ C ( s + σ )] ≥ 284 . 98 − With probability ≥ 0 . 572, C ( s + σ ) ≥ 292 − With probability ≥ 0 . 883, C ( s + σ ) ≥ 279 − With probability ≥ 0 . 987, C ( s + σ ) ≥ 268 Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal, total lat: 147 PNE, total lat.: 148 Upper: 6x13 Upper: 7x15 Middle: 1x15 Middle: 1x15 6 7 x Lower: 3x18 7 Lower: 2x14 8 x s t s t s t 10 10 10 x +1 1 1 players players players 4 x +2 3 2 3 3 x +3 Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal, total lat: 147 PNE, total lat.: 148 Upper: 6x13 Upper: 7x15 Middle: 1x15 Middle: 1x15 6 7 x Lower: 3x18 7 Lower: 2x14 8 x s t s t s t 10 10 10 x +1 1 1 players players players 4 x +2 3 2 3 3 x +3 LLF, 3 pl., total lat.: 149 ! Upper: 7x14 Middle: 0 7 Lower: 3x17 7 s t 10 0 players 3 3 Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal, total lat: 147 PNE, total lat.: 148 Upper: 6x13 Upper: 7x15 Middle: 1x15 Middle: 1x15 6 7 x Lower: 3x18 7 Lower: 2x14 8 x s t s t s t 10 10 10 x +1 1 1 players players players 4 x +2 3 2 3 3 x +3 LLF, 3 pl., total lat.: 149 ! Cover, 3 pl., total lat.: 148 Upper: 7x14 Upper: (1+6)x15 Middle: 0 Middle: (1+0)x15 2 + 6 1 + 6 7 Lower: 3x17 7 Lower: (1+1)x14 s t s t 10 10 0 1 players players 3 3 1 + 1 2 + 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Stackelberg Strategies: Examples Optimal, total lat: 147 PNE, total lat.: 148 Upper: 6x13 Upper: 7x15 Middle: 1x15 Middle: 1x15 6 7 x Lower: 3x18 7 Lower: 2x14 8 x s t s t s t 10 10 10 x +1 1 1 players players players 4 x +2 3 2 3 3 x +3 LLF, 3 pl., total lat.: 149 ! Cover, 3 pl., total lat.: 148 Upper: 7x14 Upper: (1+6)x15 Middle: 0 Middle: (1+0)x15 2 + 6 1 + 6 7 Lower: 3x17 7 Lower: (1+1)x14 s t s t 10 10 0 1 players players 3 3 1 + 1 2 + 1 Scale : I E [ C ( s + σ )] ≥ 148 . 0083 − With probability 119 120 , C ( s + σ ) = 148 1 − With probability 120 , C ( s + σ ) = 149 Dimitris Fotakis Congestion Games and Price of Anarchy
Work on Non-Atomic Games PoA for Stackelberg Routing on Parallel Links [Rough02] NP-complete to compute the best Stackelberg strategy. PoA LLF is 1 /γ for general and 4 / ( 3 + γ ) for linear latencies. Dimitris Fotakis Congestion Games and Price of Anarchy
Work on Non-Atomic Games PoA for Stackelberg Routing on Parallel Links [Rough02] NP-complete to compute the best Stackelberg strategy. PoA LLF is 1 /γ for general and 4 / ( 3 + γ ) for linear latencies. ∃ s − t networks with unbounded PoA under any strategy [BHS08] Dimitris Fotakis Congestion Games and Price of Anarchy
Work on Non-Atomic Games PoA for Stackelberg Routing on Parallel Links [Rough02] NP-complete to compute the best Stackelberg strategy. PoA LLF is 1 /γ for general and 4 / ( 3 + γ ) for linear latencies. ∃ s − t networks with unbounded PoA under any strategy [BHS08] PoA of LLF and Scale PoA LLF ≤ γ + ( 1 − γ ) α ( D ) for parallel links [Swamy07] PoA LLF ≤ 1 + 1 /γ for series-parallel networks [Sw07], [CorStMos07] PoA of LLF and Scale for linear congestion games [KarakKolliop06] Dimitris Fotakis Congestion Games and Price of Anarchy
Work on Non-Atomic Games PoA for Stackelberg Routing on Parallel Links [Rough02] NP-complete to compute the best Stackelberg strategy. PoA LLF is 1 /γ for general and 4 / ( 3 + γ ) for linear latencies. ∃ s − t networks with unbounded PoA under any strategy [BHS08] PoA of LLF and Scale PoA LLF ≤ γ + ( 1 − γ ) α ( D ) for parallel links [Swamy07] PoA LLF ≤ 1 + 1 /γ for series-parallel networks [Sw07], [CorStMos07] PoA of LLF and Scale for linear congestion games [KarakKolliop06] Non-PoA Results FPTAS for parallel links with polynomial latencies [KumMar02] Smallest fraction of coordinated players for optimality [KapSpir06] Smallest fraction of coordinated players to improve [SharWilliam07] Dimitris Fotakis Congestion Games and Price of Anarchy
Our Work on Atomic Games Games with Linear Latencies and Arbitrary Actions Upper bounds on PoA of LLF, Scale, Cover, and combinations. Nearly matching lower bound on PoA of LLF . Lower bound on PoA of any randomized Stackelberg strategy. Dimitris Fotakis Congestion Games and Price of Anarchy
Our Work on Atomic Games Games with Linear Latencies and Arbitrary Actions Upper bounds on PoA of LLF, Scale, Cover, and combinations. Nearly matching lower bound on PoA of LLF . Lower bound on PoA of any randomized Stackelberg strategy. Games on Parallel Links with Arbitrary Latencies Same upper bounds on PoA of LLF as for non-atomic games. For arbitrary latencies, PoA LLF ≤ 1 /γ For latencies in class D , PoA LLF ≤ γ + ( 1 − γ ) α ( D ) For linear latencies, PoA LLF ≤ ( 4 − γ ) / 3 . Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper Bounds Notation : Stackelberg configuration : s , s e Worst Nash equilibrium : σ , σ e Worst configuration : f = s + σ , f e = s e + σ e Approach similar to [AzarAwerEpst 05], [ChristKouts 05] . Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper Bounds Notation : Stackelberg configuration : s , s e Worst Nash equilibrium : σ , σ e Worst configuration : f = s + σ , f e = s e + σ e Approach similar to [AzarAwerEpst 05], [ChristKouts 05] . Nash inequality for selfish player i c i ( f ) ≤ � e ∈ o i ( a e ( f e + 1 ) + b e ) Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper Bounds Notation : Stackelberg configuration : s , s e Worst Nash equilibrium : σ , σ e Worst configuration : f = s + σ , f e = s e + σ e Approach similar to [AzarAwerEpst 05], [ChristKouts 05] . Nash inequality for selfish player i c i ( f ) ≤ � e ∈ o i ( a e ( f e + 1 ) + b e ) Optimal action for coordinated player j c j ( f ) = � e ∈ o j ( a e f e + b e ) Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper Bounds Notation : Stackelberg configuration : s , s e Worst Nash equilibrium : σ , σ e Worst configuration : f = s + σ , f e = s e + σ e Approach similar to [AzarAwerEpst 05], [ChristKouts 05] . Nash inequality for selfish player i c i ( f ) ≤ � e ∈ o i ( a e ( f e + 1 ) + b e ) Optimal action for coordinated player j c j ( f ) = � e ∈ o j ( a e f e + b e ) Putting everything together C ( f ) ≤ � e ∈ E ( a e ( o e f e + o e − s e ) + b e o e ) Bound rhs in terms of C ( f ) and C ( o ) using strategy’s properties. Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper Bound for LLF , 3 − 2 γ + √ 5 − 4 γ � 20 − 11 γ � PoA LLF ≤ min 8 2 2.4 2.2 Price of Anarchy of LLF 2 1.8 1.6 1.4 1.2 1 0 0.2 0.4 0.6 0.8 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper and Lower Bound for LLF , 3 − 2 γ + √ 5 − 4 γ � 20 − 11 γ � 5 ( 2 − γ ) − ε ≤ PoA LLF ≤ min 4 + γ 8 2 2.4 2.2 Price of Anarchy of LLF 2 1.8 1.6 1.4 1.2 1 0 0.2 0.4 0.6 0.8 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper and Lower Bound for Scale � 5 − 3 γ � I E [ C ( f )] , 5 − 4 γ ≤ max C ( o ) 2 3 − 2 γ 2.4 2.2 Price of Anarchy of Scale 2 1.8 1.6 1.4 1.2 1 0 0.2 0.4 0.6 0.8 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper and Lower Bound for Scale � 5 − 3 γ � I E [ C ( f )] , 5 − 4 γ ≤ max C ( o ) 2 3 − 2 γ 5 − 5 γ + 2 γ 2 − ε γ ∈ [ 0 , 1 / 2 ) 2.4 I E [ C ( f )] 2 ≥ C ( o ) 2 1 + γ − ε γ ∈ [ 1 / 2 , 1 ] 2.2 Price of Anarchy of Scale 2 1.8 1.6 1.4 1.2 1 0 0.2 0.4 0.6 0.8 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper Bound for Cover PoA of Cover tends to PoA of non-atomic game as λ grows Linear latencies : PoA Cover ≤ 4 λ − 1 3 λ − 1 1 Linear latencies without offset : PoA Cover ≤ 1 + 2 λ 1 Parallel links, linear latencies, no offset : PoA Cover ≤ 1 + 4 ( λ + 1 ) 2 − 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper Bound for Cover-Scale If n / m large and k ≥ m , Cover-Scale achieves better PoA ! Cover assigns λ m players so that min { λ, o e } ≤ s C e ≤ o e for each e , λ ≤⌊ k / m ⌋ any integer. Scale assigns k − λ m players randomly wrt o − s C . n / m = 10 2.4 k ≥ m Price of Anarchy of Cover-Scale γ = k / n ≥ 0 . 1 2.2 λ = 1 − − Scale 2 − − Cover-Scale 1.8 1.6 1.4 1.2 1 0.2 0.4 0.6 0.8 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Linear Games: Upper Bound for LLF-Cover If n / m large and k ≥ m , LLF-Cover achieves better PoA ! LLF assigns k − λ m players to the largest latency actions in o . Cover assigns λ m players so that min { λ, o e − s L e } ≤ s C e ≤ o e − s L e for each e , λ ≤⌊ k / m ⌋ any integer. n / m = 10 2.4 k ≥ m Price of Anarchy of LLF-Cover γ = k / n ≥ 0 . 1 2.2 λ = 1 − − LLF 2 − − LLF-Cover 1.8 1.6 1.4 1.2 1 0.2 0.4 0.6 0.8 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls Economic (dis)incentives (tolls) to improve total latency . Players minimize latency + tolls and reach better PNE. Objective : moderate and efficiently computable tolls leading players to PNE of optimal total latency. Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls Economic (dis)incentives (tolls) to improve total latency . Players minimize latency + tolls and reach better PNE. Objective : moderate and efficiently computable tolls leading players to PNE of optimal total latency. Symmetric network congestion game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) Toll function τ : E �→ I R ≥ 0 assigns toll τ e ≥ 0 to every edge e . Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls Economic (dis)incentives (tolls) to improve total latency . Players minimize latency + tolls and reach better PNE. Objective : moderate and efficiently computable tolls leading players to PNE of optimal total latency. Symmetric network congestion game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) Toll function τ : E �→ I R ≥ 0 assigns toll τ e ≥ 0 to every edge e . Modified congestion game with tolls Γ τ ( N , G ( V , E ) , ( d e ) e ∈ E ) Cost of edge e in configuration σ : d e ( σ e ) = d e ( σ e ) + τ e Cost of player i in configuration σ : c i ( σ ) = � e ∈ σ i ( d e ( σ e ) + τ e ) Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls Economic (dis)incentives (tolls) to improve total latency . Players minimize latency + tolls and reach better PNE. Objective : moderate and efficiently computable tolls leading players to PNE of optimal total latency. Symmetric network congestion game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) Toll function τ : E �→ I R ≥ 0 assigns toll τ e ≥ 0 to every edge e . Modified congestion game with tolls Γ τ ( N , G ( V , E ) , ( d e ) e ∈ E ) Cost of edge e in configuration σ : d e ( σ e ) = d e ( σ e ) + τ e Cost of player i in configuration σ : c i ( σ ) = � e ∈ σ i ( d e ( σ e ) + τ e ) Refundable tolls increase players’ cost but not total latency Admin seeks to minimize total latency C ( σ ) = � e ∈ E σ e d e ( σ e ) Tolls τ such that optimal o of Γ is some (the worst) PNE of Γ τ Dimitris Fotakis Congestion Games and Price of Anarchy
Enforceable Congestions Configuration f weakly enforceable by tolls τ Every configuration σ with σ e = f e on all e ∈ E is a PNE of Γ τ Weakly optimal tolls τ weakly enforce optimal o : PoS = 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Enforceable Congestions Configuration f weakly enforceable by tolls τ Every configuration σ with σ e = f e on all e ∈ E is a PNE of Γ τ Weakly optimal tolls τ weakly enforce optimal o : PoS = 1 Configuration f strongly enforceable by tolls τ Configuration σ is a PNE of Γ τ iff σ e = f e for all e ∈ E Strongly enforceable : weakly enforceable and unique PNE of Γ τ (Strongly) optimal tolls τ strongly enforce optimal o : PoA = 1 Dimitris Fotakis Congestion Games and Price of Anarchy
Enforceable Congestions Configuration f weakly enforceable by tolls τ Every configuration σ with σ e = f e on all e ∈ E is a PNE of Γ τ Weakly optimal tolls τ weakly enforce optimal o : PoS = 1 Configuration f strongly enforceable by tolls τ Configuration σ is a PNE of Γ τ iff σ e = f e for all e ∈ E Strongly enforceable : weakly enforceable and unique PNE of Γ τ (Strongly) optimal tolls τ strongly enforce optimal o : PoA = 1 Weak and strong optimality equivalent for non-atomic games. Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls: Example Optimal PNE Total lat.: 147 Total lat.: 148 6 7 7 8 x x s t t s t 10 s 10 10 1 1 x +1 players players players 3 4 2 3 x +2 3 x +3 Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls: Example Optimal PNE Total lat.: 147 Total lat.: 148 6 7 7 8 x x s t t s t 10 s 10 10 1 1 x +1 players players players 3 4 2 3 x +2 3 x +3 Total lat.: 147 Upper: 6x(13+3) Middle: 1x(15+2) 6 Lower: 3x(18+0) 7 x x + 3 t t s s 10 10 1 x +1+ 2 players players x +2 4 3 x +3 3 Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls: Example Optimal PNE Total lat.: 147 Total lat.: 148 6 7 7 8 x x s t t s t 10 s 10 10 1 1 x +1 players players players 3 4 2 3 x +2 3 x +3 Upper: 7x(15+3) Middle: 1x(15+2) Lower: 2x(14+0) Optimal Total lat.: 147 Total lat.: 149 weakly enforceable Upper: 6x(13+3) Upper: 7x(14+3) Middle: 1x(15+2) Middle: 0 6 7 Lower: 3x(18+0) 7 Lower: 3x(17+0) 7 x x + 3 t t t s s s 10 10 10 1 0 x +1+ 2 players players players x +2 4 3 3 x +3 3 3 Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls: Example Optimal PNE Total lat.: 147 Total lat.: 148 6 7 7 8 x x s t t s t 10 s 10 10 1 1 x +1 players players players 3 4 2 3 x +2 3 x +3 Upper: 7x(15+3) Middle: 1x(15+2) Lower: 2x(14+0) Optimal Total lat.: 147 Total lat.: 149 weakly enforceable Upper: 6x(13+3) Upper: 7x(14+3) Middle: 1x(15+2) Middle: 0 6 7 Lower: 3x(18+0) 7 Lower: 3x(17+0) 7 x x + 3 t t t s s s 10 10 10 1 0 x +1+ 2 players players players x +2 4 3 3 x +3 3 3 Optimal Total lat.: 147 strongly enforceable Upper: 6x(13+5) Middle: 1x(15+3) 6 Lower: 3x(18+0) 7 x x + 5 s t s t 10 10 1 x +1+ 3 players players 4 x +2 3 3 x +3 Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls for Non-Atomic Games Marginal cost tolls for optimal : d e ( o e ) = d e ( o e ) + o e d ′ e ( o e ) Optimal of Γ iff Nash equilibrium of Γ τ Not weakly enforce optimal for atomic games. Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls for Non-Atomic Games Marginal cost tolls for optimal : d e ( o e ) = d e ( o e ) + o e d ′ e ( o e ) Optimal of Γ iff Nash equilibrium of Γ τ Not weakly enforce optimal for atomic games. Optimal tolls for heterogeneous players Players have different latency vs. tolls valuation. Existence for s − t networks and computation for finite #types [ColeDodisRough 03] Existence of moderate tolls for s − t networks and computation for series-parallel nets and infinite #types [Flei 04] Efficient computation follows from LP duality [FJM 04], [KaraKoll 04] Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls for Atomic Games Homogeneous players, linear latencies [CaragKaklKanel 06] Simple non-symmetric game with PoA ≥ 1 . 2 for any tolls. ∃ non-symmetric games not admitting optimal tolls! Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls for Atomic Games Homogeneous players, linear latencies [CaragKaklKanel 06] Simple non-symmetric game with PoA ≥ 1 . 2 for any tolls. ∃ non-symmetric games not admitting optimal tolls! Simple optimal tolls for parallel links Efficiently computable tolls reducing PoA to 2 + ε Not strongly optimal for series-parallel networks. Dimitris Fotakis Congestion Games and Price of Anarchy
Refundable Tolls for Atomic Games Homogeneous players, linear latencies [CaragKaklKanel 06] Simple non-symmetric game with PoA ≥ 1 . 2 for any tolls. ∃ non-symmetric games not admitting optimal tolls! Simple optimal tolls for parallel links Efficiently computable tolls reducing PoA to 2 + ε Not strongly optimal for series-parallel networks. Optimal tolls for s − t networks other than parallel links? Dimitris Fotakis Congestion Games and Price of Anarchy
Cost-Balancing Tolls Cost-balancing tolls τ for f ∀ p ∈ P with min e ∈ p { f e } > 0 and ∀ p ′ ∈ P , � � ( d e ( f e ) + τ e ) ≤ ( d e ( f e ) + τ e ) e ∈ p e ∈ p ′ Any used path becomes min-cost path. Dimitris Fotakis Congestion Games and Price of Anarchy
Cost-Balancing Tolls Cost-balancing tolls τ for f ∀ p ∈ P with min e ∈ p { f e } > 0 and ∀ p ′ ∈ P , � � ( d e ( f e ) + τ e ) ≤ ( d e ( f e ) + τ e ) e ∈ p e ∈ p ′ Any used path becomes min-cost path. Any configuration σ with σ e = f e for all e ∈ E is a PNE of Γ τ Configuration f weakly enforceable by cost-balancing tolls for it. Dimitris Fotakis Congestion Games and Price of Anarchy
Cost-Balancing Tolls Cost-balancing tolls τ for f ∀ p ∈ P with min e ∈ p { f e } > 0 and ∀ p ′ ∈ P , � � ( d e ( f e ) + τ e ) ≤ ( d e ( f e ) + τ e ) e ∈ p e ∈ p ′ Any used path becomes min-cost path. Any configuration σ with σ e = f e for all e ∈ E is a PNE of Γ τ Configuration f weakly enforceable by cost-balancing tolls for it. Which configurations admit cost-balancing tolls? Dimitris Fotakis Congestion Games and Price of Anarchy
Our Results on Cost-Balancing Tolls Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Dimitris Fotakis Congestion Games and Price of Anarchy
Our Results on Cost-Balancing Tolls Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Moderate : tolls paid by any player in o ≤ max latency in o Dimitris Fotakis Congestion Games and Price of Anarchy
Our Results on Cost-Balancing Tolls Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Moderate : tolls paid by any player in o ≤ max latency in o Strongly optimal for series-parallel networks. ∃ networks where not strongly optimal. Dimitris Fotakis Congestion Games and Price of Anarchy
Our Results on Cost-Balancing Tolls Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Moderate : tolls paid by any player in o ≤ max latency in o Strongly optimal for series-parallel networks. ∃ networks where not strongly optimal. Heterogenous players : ∃ parallel-link games not admitting strongly optimal tolls. Dimitris Fotakis Congestion Games and Price of Anarchy
Our Results on Cost-Balancing Tolls Cost-balancing tolls Computable in linear time for (acyclic) optimal configuration o Weakly optimal for s − t networks (for heterogeneous players!). Moderate : tolls paid by any player in o ≤ max latency in o Strongly optimal for series-parallel networks. ∃ networks where not strongly optimal. Heterogenous players : ∃ parallel-link games not admitting strongly optimal tolls. Complexity of computing best optimal tolls NP-hard even for linear games on series-parallel networks. For 2-player linear games on series-parallel networks, NP-hard to distinguish between PoA = 1 and PoA ≥ 1 . 2. Dimitris Fotakis Congestion Games and Price of Anarchy
Computing Cost-Balancing Tolls Game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) and acyclic optimal o . 8 x 0 0 8 x 2 x x x 3 x x x x 5 x 2 x players x x s t 5 Dimitris Fotakis Congestion Games and Price of Anarchy
Computing Cost-Balancing Tolls Game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) and acyclic optimal o . E o = { e ∈ E : o e > 0 } : G ( V , E o ) is directed acyclic graph ( DAG ) ∀ edge e ∈ E o , edge length ℓ e = d e ( o e ) 8 x 0 0 8 x 2 x x x 3 x x x x 5 x 2 x players x x s t 5 Dimitris Fotakis Congestion Games and Price of Anarchy
Computing Cost-Balancing Tolls Game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) and acyclic optimal o . E o = { e ∈ E : o e > 0 } : G ( V , E o ) is directed acyclic graph ( DAG ) ∀ edge e ∈ E o , edge length ℓ e = d e ( o e ) 8 0 0 1 1 2 1 1 1 1 2 s t Dimitris Fotakis Congestion Games and Price of Anarchy
Computing Cost-Balancing Tolls Game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) and acyclic optimal o . E o = { e ∈ E : o e > 0 } : G ( V , E o ) is directed acyclic graph ( DAG ) ∀ edge e ∈ E o , edge length ℓ e = d e ( o e ) Longest path tree from s in linear time ∀ vertex u , ℓ u = length of longest s − u path in G o 8 1 1 4 4 0 0 1 1 2 0 1 8 1 1 1 2 s 3 t 1 4 Dimitris Fotakis Congestion Games and Price of Anarchy
Computing Cost-Balancing Tolls Game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) and acyclic optimal o . E o = { e ∈ E : o e > 0 } : G ( V , E o ) is directed acyclic graph ( DAG ) ∀ edge e ∈ E o , edge length ℓ e = d e ( o e ) Longest path tree from s in linear time ∀ vertex u , ℓ u = length of longest s − u path in G o ∀ e = ( u , v ) ∈ E o , τ e = ℓ v − ( ℓ u + d e ( o e )) ∀ e �∈ E o , τ e = τ max ≥ ℓ t 8 0 1 1 4 4 0 0 0 0 0 3 0 0 1 1 2 0 1 8 τ max τ max 1 1 1 2 s 3 t 0 1 0 2 1 4 Dimitris Fotakis Congestion Games and Price of Anarchy
Computing Cost-Balancing Tolls Game Γ( N , G ( V , E ) , ( d e ) e ∈ E ) and acyclic optimal o . E o = { e ∈ E : o e > 0 } : G ( V , E o ) is directed acyclic graph ( DAG ) ∀ edge e ∈ E o , edge length ℓ e = d e ( o e ) Longest path tree from s in linear time ∀ vertex u , ℓ u = length of longest s − u path in G o ∀ e = ( u , v ) ∈ E o , τ e = ℓ v − ( ℓ u + d e ( o e )) ∀ e �∈ E o , τ e = τ max ≥ ℓ t Non-negative tolls ∀ e = ( u , v ) ∈ E o , ℓ v ≥ ℓ u + d e ( o e ) 8 0 1 1 4 4 0 0 0 0 0 3 0 0 1 1 2 0 1 8 τ max τ max 1 1 1 2 s 3 t 0 1 0 2 1 4 Dimitris Fotakis Congestion Games and Price of Anarchy
Computing Cost-Balancing Tolls ∀ e = ( u , v ) ∈ E o , τ e = ℓ v − ( ℓ u + d e ( o e )) ∀ e �∈ E o , τ e = τ max ≥ ℓ t Cost-balancing tolls for o ∀ used p ∈ P : � e ∈ p ( d e ( o e ) + τ e ) = � e ∈ p ( ℓ v ( e ) − ℓ u ( e ) ) = ℓ t Otherwise, ∃ e ∈ p \ E o with τ e = τ max ≥ ℓ t 8 0 1 1 4 4 0 0 0 0 0 3 0 0 1 1 2 1 0 8 τ max τ max 1 1 1 2 s 3 t 0 1 0 2 1 4 Dimitris Fotakis Congestion Games and Price of Anarchy
Computing Cost-Balancing Tolls ∀ e = ( u , v ) ∈ E o , τ e = ℓ v − ( ℓ u + d e ( o e )) ∀ e �∈ E o , τ e = τ max ≥ ℓ t Cost-balancing tolls for o ∀ used p ∈ P : � e ∈ p ( d e ( o e ) + τ e ) = � e ∈ p ( ℓ v ( e ) − ℓ u ( e ) ) = ℓ t Otherwise, ∃ e ∈ p \ E o with τ e = τ max ≥ ℓ t Moderate tolls Amount of tolls paid by any player in o ≤ ℓ t Sufficiently large τ max : tolls paid by any player in any PNE ≤ ℓ t 8 0 1 1 4 4 0 0 0 0 0 3 0 0 1 1 2 1 0 8 τ max τ max 1 1 1 2 s 3 t 0 1 0 2 1 4 Dimitris Fotakis Congestion Games and Price of Anarchy
Social Ignorance in Congestion Games Motivation Players have partial information which depends on their social context. How does the social context affect inefficiency ? Dimitris Fotakis Congestion Games and Price of Anarchy
Social Ignorance in Congestion Games Motivation Players have partial information which depends on their social context. How does the social context affect inefficiency ? Ideas from Previous Work A Bayesian approach to load balancing [GairMonTiem 08] Social graph : player knows neighbors’ weights and probability distribution for others’ [KoutsPanaSpir 07] Some players are ignorant of the presence of others [KarKimViglXia 07] Social context affects individual costs [AshlKrysTennen 08] Dimitris Fotakis Congestion Games and Price of Anarchy
Graphical Congestion Games Graphical Congestion Games [BiloFaneFlamMosca 08] Social graph G = ( N , R ) . Each player / vertex has: Full information about his social neighbors. No information whatsoever about the remaining players. Players select strategies based on presumed costs. Dimitris Fotakis Congestion Games and Price of Anarchy
Graphical Congestion Games Graphical Congestion Games [BiloFaneFlamMosca 08] Social graph G = ( N , R ) . Each player / vertex has: Full information about his social neighbors. No information whatsoever about the remaining players. Players select strategies based on presumed costs. Consequences Linear graphical games admit potential function (and PNE). PoA ≤ n (∆( G ) + 1 ) PoS ≤ n Dimitris Fotakis Congestion Games and Price of Anarchy
Our Work on Graphical Congestion Games Question Which parameter of the social graph characterizes inefficiency of PNE and the Nash dynamics (for weighted players too) ? Dimitris Fotakis Congestion Games and Price of Anarchy
Our Work on Graphical Congestion Games Question Which parameter of the social graph characterizes inefficiency of PNE and the Nash dynamics (for weighted players too) ? Our Answer The independence number α ( G ) of the social graph G : � 3 α ( G )+ 7 3 α ( G )+ 1 α 2 ( G ) if α ( G ) < n 2 , PoA ≤ if α ( G ) ≥ n 2 n ( n − α ( G ) + 1 ) 2 α ( G ) ≤ PoS ≤ 2 α ( G ) Convergence to PNE not slower due to social ignorance (what about faster?). Dimitris Fotakis Congestion Games and Price of Anarchy
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